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arXiv · 2209.07283

On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$

Maxim Kirsebom·Keivan Mallahi-Karai
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Abstract

We study an extreme value distribution for the unipotent flow on the modular surface $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for $F(r)$ for $r \in [-\frac{1}{2} \log 2, \infty)$, and establish asymptotic behavior of $F(r)$ as $r \to -\infty$.

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Maxim Kirsebom, Keivan Mallahi-Karai. 2022-08-23. On an extreme value law for the unipotent flow on $\mathrm{SL}_2(\mathbb{R})/\mathrm{SL}_2(\mathbb{Z})$. https://arxiv.org/abs/2209.07283

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